Proposition 3.9 
There exist spined categories \((\mathcal {C}, \Omega , \mathfrak {P})\)  satisfying \(|\Omega _n| = n = \omega (\Omega _n)\)  that do not admit any S-functors.

Consider the category \(\mathbf {Poset}_{mono}\)  which has finite posets as objects and order-preserving injections as morphisms. Let \(\Omega _n\)  denote set \(\left\lbrace  {m \in \mathbb {N}}\:|\:{m \le n} \right\rbrace \)  under its usual linear ordering, and let \(\mathfrak {P}\)  assign to each span of the form

G  n [l, "g"'] [r, "h"]  H

the pushout \(G +_{\Omega _n} H\)  of the span in \(\mathbf {Poset}_{homo}\)  (the category of posets is cocomplete [1]}, so in particular it has all pushouts). We will show that the structure \((\mathbf {Poset}_{mono}, \Omega , \mathfrak {P})\)  forms a spined category that does not admit any S-functors.
